2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Geometry H
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Hisaaki Endo
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Fri
- Class
- -
- Course Code
- MTH.B504
- Number of credits
- 100
- Course offered
- 2024
- Offered quarter
- 4Q
- Syllabus updated
- Mar 14, 2025
- Language
- English
Syllabus
Course overview and goals
The aim of this lecture is to familiarize the students with the basic language of and some fundamental theorems for mapping class groups of surfaces. This course is a continuation of [MTH.B503 : Advanced topics in Geometry G].
Course description and aims
Students are expected to
・understand proofs of fundamental theorems on Lefschetz fibrations.
Keywords
Charts, fiber sums, stabilizations, Kirby diagrams, monodromy substitutions, rational blowdowns, holomorphicity, Stein surfaces.
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | General theory of charts | Details will be provided in class. |
| Class 2 | Stabilizations of Lefschetz fibrations | Details will be provided in class. |
| Class 3 | Fiber sums and hyperellipticity | Details will be provided in class. |
| Class 4 | Kirby diagrams and Kirby claculus | Details will be provided in class. |
| Class 5 | Monodromy substitutions and rational blowdowns | Details will be provided in class. |
| Class 6 | Relative invariants and holomorphicity | Details will be provided in class. |
| Class 7 | PALFs and Stein surfaces | Details will be provided in class. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None
Reference books, course materials, etc.
R. I. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus, American Mathematical Society, 1999.
H. Endo and K. Hayano, 4-manifolds and fibrations, in Japanese, Kyoritsu Shuppan, 2024.
Evaluation methods and criteria
Homework assignments (100％)
Related courses
- MTH.B202 ： Introduction to Topology II
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B341 ： Topology
- MTH.B503 ： Advanced topics in Geometry G
Prerequisites
Basic algebraic topology (homology, cohomology, and the fundamental group), smooth manifolds, and the previous quarter of this class.
Other
To be announced.